1. Explain with the help of a suitable example, what do you understand by linear programming.

2. What are the characteristics and limitations of a linear programming problem?

3. What do you understand by graphical method? Give its limitations.

- A linear programming problem has a well defined objective function which is ...... and which is to be ...... or .......
- The constraints in a linear programming problem arises due to limitation of ...... These are linear ...... or ......
- The solution of a linear programming problem indicates the right combination of ...... which ...... or ...... the objective function satisfying the various .......

1. Jumpin Ltd. has canned apple and bottled juice as its products with profit margin Rs. 2 and Rs. 1 respectively per unit. The following table indicates the labour, equipment and material to produce each product per unit.

Bottled Juice | Canned Apple | Total | |

Labour (man hours) | 3 | 2 | 12 |

Equipment (machine hours) | 1 | 2.3 | 6.9 |

Material (unit) | 1 | 1.4 | 4.9 |

Formulate the problem specifying the product mix, which will maximize profit without exceeding the various levels of resources.

2. The managing director of a small scale company decides to manufacture two products, P1 & P2, each of which is processed in two shops, viz. Machining shop (M) and Finishing shop (F). One unit of P1 takes 15 hours of machining and 24 hours of finishing shops. The corresponding requirements for P2 are 25 hours and 11 hours respectively in shops. The total available hours per day in M and F shops are 375 and 264 respectively. P1 gives a profit of Rs. 18 per unit and P2 Rs. 16 per unit. Formulate the above problem.

3. A company owns two flour mills, A and B, which have different production capacities for high, medium and flour. This company has entered a contract to supply flour to a firm every week with at least 12, 8 and 24 quintals of high, medium and low grade respectively. It costs the company Rs. 1000 and Rs. 800 per day to run mill A and B respectively. On a day, mill A produce 6, 2 and 4 quintals of high, medium and low grade flour respectively. Mill B produce 2, 2 and 12 quintals of high, medium and low grade flour respectively. How many days per week should each mill be operated in order to meet the contract order most economically.

4. Decibel Electronics produces two products A and B that are sold on a weekly basis. The weekly production cannot exceed 25 for product A and 35 for product B. The company employs a total of 80 workers. Product A requires 2 man-weeks of labour whereas B requires only 1. A gives a profit of Rs. 16 and B Rs. 40. Formulate the above LPP.

5. A company that produces soft drinks has a contract that requires
that a minimum of 80 units of the chemical A and 60 units of the chemical
B go into each bottle of the drink. The chemicals are available in a
prepared mix from two different suppliers. Supplier X_{1} has
a mix of 4 units of A and 2 units of B that costs Rs. 10, and supplier
X_{2} has a mix of 1 units of A and 1 unit of B that costs Rs.
4. How many mixes from company X_{1} and company X_{1} should the company purchase to honour contract requirement and yet minimize
cost?

1. Maximize: z = 3x + 2y

subject to the constraints:

x - y ≤ 1

x + y ≥ 3

x, y ≥ 0

2. Maximize: z = x + y

subject to the constraints:

x + y ≤ 1

-3x + y ≥ 3

x, y ≥ 0

3. A manufacture of packing material, manufacturers two type of packing tins, round and flat. Major production facilities involved are cutting and joining. The cutting department can process 300 round tins or 500 flat tins per hour. The joining department can process 500 round tins or 300 flat tins per hour. If the contribution towards profit for around tin is the same as that of a flat tin what is that the optimum production level?

4. Maximize z = 3x_{1} - 4x_{2}

subject to

x_{1} - x_{2} ≥
0

x_{2 }≤ 6

x_{1}, x_{2} ≥ 0

5. Maximize z = 3x_{1} + 2x_{2}

subject to

x_{1} + 2x_{2} ≤
2

2x_{1} + x_{2 }≥ 6

x_{1}, x_{2} ≥ 0

6. Maximize z = 4x_{1} + 3x_{2}

subject to

2x_{1} + 3x_{2} ≤
6

4x_{1} + 6x_{2} ≥ 24

x_{1}, x_{2} ≥ 0

7. Maximize z = 3x_{1} + 2x_{2}

subject to

2x_{1} - 3x_{2} _{≥} 0

3x_{1} + 4x_{2 }≤ -12

x_{1}, x_{2} ≥ 0

8. Maximize z = 50x_{1} + 60x_{2}

subject to

2x_{1} + x_{2} ≤
300

3x_{1} + 4x_{2} ≤ 509

4x_{1} + 7x_{2} ≤ 812

x_{1}, x_{2} ≥ 0

9. Minimize z = 2x_{1} + 1.7x_{2}

subject to

0.15x_{1} + 0.10x_{2} ≥
1.0

0.75x_{1} + 1.70x_{2 } ≥
7.5

1.30x_{1} + 1.10x_{2} ≥
10.0

x_{1}, x_{2} ≥ 0

10. Maximize z = 3x_{1} + 2x_{2}

subject to

2x_{1} - x_{2} ≥
2

x_{1} + 2x_{2 }≤ 8

x_{1}, x_{2} ≥ 0

11. Minimize z = x + y

subject to

2x + y ≥ 12

5x + 8y ≥ 74

x + 6y ≥ 24

x, y ≥ 0

12. Minimize z = 2x_{1} + 3x_{2}

subject to

-x_{1} + 2x_{2} ≤
4

x_{1} + x_{2 }≤ 6

x_{1} + 3x_{2 }≥ 9

x_{1}, x_{2} ≥ 0

subject to

5x + 8y ≤ 2000

3x + 10y ≤ 1000

x, y ≥ 0

14. Minimize z = 7x_{1} + 3x_{2}

subject to

x_{1} + 2x_{2} ≥
3

x_{1} + x_{2 }≤ 4

0 ≤ x_{1} ≤
15/2

0 ≤ x_{2} ≤
3/2

15. Minimize z = 4x_{1} + 3x_{2}

subject to

2x_{1} + 3x_{2} ≤ 6

2x_{2 }≤ 5

-3x_{1} + 2x_{2} ≤ 3

2x_{1} + x_{2} ≤ 4

x_{1}, x_{2} ≥ 0